Implication Details
Claim: A category is abelian if and only if it is additive and has cokernels and is conormal and has kernels and is normal.
Proof: This holds by definition.
Show 71 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of abelian groups
- category of finitely generated abelian groups
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- category of Banach spaces with linear contractions
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- category of finite sets and injections
- category of finite sets and surjections
- category of filtered vector spaces
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of fields
- category of free abelian groups
- category of groups
- category of countable groups
- category of Jónsson-Tarski algebras
- category of M-sets
- category of smooth manifolds
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of sets and relations
- category of schemes
- category of sequences of abelian groups
- category of sets
- category of pointed sets
- category of set functions and commutative squares
- category of countable sets
- category of sets with finite-to-one maps
- category of pairs of sets
- category of sheaves
- category of abelian sheaves
- category of combinatorial species
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of uniform spaces
- category of vector spaces
- category of Z-functors
- preordered set of integers w.r.t. divisibility
- category of graded abelian groups
- category of graded modules over a graded ring
- real interval [0,1]
- category of simplicial sets
- walking commutative square
- walking composable pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting